The present investigation is mainly motivated by the modeling of some aspects of the random nature of fluid turbulence [1]}, [2]}. To be more precise, let us begin with illustrating Kolmogorov's phenomenological theory of three-dimensional turbulence [3]}. To do so, consider a component \(u_{i\in \lbrace 1,2,3\rbrace }(t,x)\) , and \(x\in \mathbb {R}^3\) , of the divergence-free vector velocity field \(u\)  of a fluid of viscosity \(\nu \) , whose dynamics is governed by the incompressible three-dimensional Navier-Stokes equations. Moreover, we assume that this evolution is supplemented by an additive random vector forcing term \(f(t,x)\)  that we take divergence-free and smooth in space. Typically, without loss of generality, and to fix ideas, consider a zero-average, white-in-time Gaussian vector field whose covariance is of the form \(\mathbb {E}[f(t,x)\cdot f(t^{\prime },x^{\prime })]\propto \delta (t-t^{\prime })\mathcal {C}_f(|x-x^{\prime }|)\) , where the scalar positive-definite function \(\mathcal {C}_f(x)\)  is \(\mathcal {C}^{\infty }\)  and takes significant values only for \(|x|\leqslant L\) .
